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Skip the Equations: Learning Behavior of Personalized Dynamical Systems Directly From Data
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Extends direct semantic modeling (Semantic ODE) to multi-dimensional, personalized dynamical systems via EPISODE, which predicts trajectory compositions and properties directly from static features using a decision tree and GAM-based property maps.
Problem
Modeling personalized (multi-dimensional, covariate-dependent) dynamical systems in a way that exposes interpretable behavior and allows editing, avoiding the infeasible two-step process of ODE discovery and post-hoc mathematical analysis.
Approach
Predict the semantic representation (shape composition and shape properties) of each trajectory dimension directly from static features: a small classification decision tree selects the composition, and per-composition property maps built from GAMs with monotonic/positivity transformations predict transition points and derivatives, so the model is inspectable and editable without deriving equation
Key results
- On the real Tacrolimus dataset, EPISODE achieves 0.255 error versus 0.337 for SINDy-5 and 0.351 for the expert PopPK model, and slightly beats NeuralODE (0.263).
- EPISODE or EPISODE* achieves the best result on most of the six datasets in Table 2, e.g., HIV at sigma=0.01 with 0.088 vs 0.175 for SINDy-5.
- After adding a penalty on the horizontal asymptote, the learned model can be set to decay to zero with no significant performance drop, yielding a biologically plausible verified model.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a constrained composition library (ablation) | EPISODE · SIR σ = 0.01 | error (Eq. 5): 0.026(.004) -> 0.012(.000) | Helped |
| Use a constrained composition library (ablation) | EPISODE · HIV σ = 0.01 | error (Eq. 5): 0.088(.006) -> 0.089(.006) | No clear effect |
| Use a constrained composition library (ablation) | EPISODE · PK σ = 0.1 | error (Eq. 5): 0.132(.013) -> 0.146(.037) | Hurt |
| Use a constrained composition library (ablation) | EPISODE · Tumor σ = 0.1 | error (Eq. 5): 0.290(.147) -> 0.165(.016) | Helped |
| Use a constrained composition library (ablation) | EPISODE · Tacrolimus (real) | error (Eq. 5): 0.256(.007) -> 0.256(.007) | No clear effect |
| Use a constrained composition library (ablation) | EPISODE · Bike sharing (real) | error (Eq. 5): 0.192(.013) -> 0.195(.029) | No clear effect |
| Enforce zero horizontal asymptote after refitting with a penalty (ablation) | EPISODE on Tacrolimus · Tacrolimus test set | error (Eq. 5): 0.255(.113) -> 0.257(.109) | No clear effect |
| Use EPISODE rather than ANODE | Tacrolimus forecasting · Tacrolimus test set | error (Eq. 5): 0.268(.113) -> 0.255(.113) | Helped |
| Use EPISODE rather than the expert PopPK model | Tacrolimus forecasting · Tacrolimus test set | error (Eq. 5): 0.351(.177) -> 0.255(.113) | Helped |
Surprising findings
- EPISODE can outperform black-box approaches (NeuralODE, LatentODE) on some settings, which the authors attribute to inductive biases acting as regularizers.
Limitations
- Restricted to finite compositions, so oscillatory or periodic behavior cannot be modeled.
- Fitting the composition map is time-consuming because every composition must be matched to each sample.
- Some datasets show high variance and some WSINDy runs fail to converge; performance on the real Bike sharing dataset is weaker than black-box baselines.
Experimental setup
Datasets: Tacrolimus (real PopPK) SIR (synthetic) HIV PK (synthetic) Tumor (synthetic) Bike sharing
Benchmarks: Tacrolimus SIR HIV PK Tumor Bike sharing
Metrics: normalized RMSE over trajectories (Equation 5)
Baselines: NeuralODE ANODE LatentODE SINDy-5 SINDy-20 WSINDy-5 WSINDy-20 Expert PopPK model
Builds on: Semantic ODE SINDy WSINDy Neural ODE Latent ODE
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